Non-linear and hysteretical finite element formulation applied to magnetostrictive materials

The final publication is available at link.springer.com.[EN] Giant magnetostrictive actuators are suitable for applications requiring large mechanical displacements under low magnetic fields; for instance Terfenol-D made out of rare earth-iron materials can produce important strains. But these actuators exhibit hysteretic non-linear behavior, making it very difficult to experimentally characterize them. Therefore, sophisticated numerical algorithms to develop computational tools are necessary. In this work, theoretical and numerical formulations within the finite element method are developed to simulate magnetostriction. Theoretically, within the framework of non-equilibrium thermodynamics, the hysteresis is introduced by the Debye-memory relaxation. Numerically, the main novelty is the time integration, coupled Newmark-beta (for mechanical) and convolution integrals (for magnetic constitutive equations); the non-linearity is solved with the standard Newton-Raphson algorithm. Constitutive non-linearities are incorporated with the Maxwell stress tensor, quadratically dependent on the magnetic field. The numerical code is validated using analytical and experimental solutions; several examples are presented to demonstrate the capabilities of the present formulation.Palma, R.; Pérez-Aparicio, JL.; Taylor, RL. (2020). Non-linear and hysteretical finite element formulation applied to magnetostrictive materials. Computational Mechanics. 65(6):1433-1445. https://doi.org/10.1007/s00466-020-01828-yS14331445656Orszulik RR, Gabbert U (2015) An interface between Abaqus and Simulink for high-fidelity simulations of smart structures. IEEE/ASME Trans Mechatron 21(2):879–887Melchor J, Rus G (2014) Torsional ultrasonic transducer computational design optimization. Ultrasonics 54(7):1950–1962Khan FU, Ahmad I (2016) Review of energy harvesters utilizing bridge vibrations. Shock Vib 2016:1–21Melingui A, Lakhal O, Daachi B, Bosco J, Merzouki R (2015) Adaptive neural network control of a compact bionic handling arm. IEEE/ASME Trans Mechatron 20(6):2862–2875Anjanappa M, Bi J (1994) A theoretical and experimental study of magnetostrictive mini-actuators. Smart Mater Struct 3:83–91Anjanappa M, Bi J (1994) Magnetostrictive mini actuators for smart structure applications. Smart Mater Struct 3:383–390Venkataraman R, Rameau J, Krishnaprasad PS. Characterization of an ETREMA MP 50/6 magnetostrictive actuator, TR 98-1. Technical report of the Institute for System Research, University of Maryland at College ParkLi Z, Zhang X, Gu GY, Chen X, Su CY (2016) A comprehensive dynamic model for magnetostrictive actuators considering different input frequencies with mechanical loads. IEEE Trans Ind Inform 12(3):980–990Pérez-Aparicio JL, Sosa H (2004) A continuum three-dimensional, fully coupled, dynamic, non-linear finite element formulation for magnetostrictive materials. Smart Mater Struct 13:493–502Perez-Aparicio JL, Palma R, Taylor RL (2016) Multiphysics and thermodynamic formulations for equilibrium and non-equilibrium interactions: non-linear finite elements applied to multi-coupled active materials. Arch Comput Methods Engieering 23(3):535–583Kannan KS, Dasgupta A (1997) A non-linear Galerkin finite-element theory for modeling magnetostrictive smart structures. Smart Mater Struct 6:341–350Kiang J, Tong L (2010) Nonlinear magneto-mechanical finite element analysis of Ni–Mn–Ga single crystals. Smart Mater Struct 19:1–17Zhou P (2015) On the coupling effects between elastic and electromagnetic fields from the perspective of conservation of energy. arXiv:1512.04487Natale C, Velardi F, Visone C (2001) Identification and compensation of Preisach hysteresis models for magnetostrictive actuators. Phys B Condens Matter 306(1):161–165Kaltenbacher M, Meiler M, Ertl M (2009) Physical modeling and numerical computation of magnetostriction. Int J Comput Math Electr Electron Eng 28(4):819–832Linnemann K, Klinkel S, Wagner W (2009) A constitutive model for magnetostrictive and piezoelectric materials. Int J Solids Struct 46:1149–1166de Groot SR, Mazur P (1984) Non-equilibrium thermodynamics. Dover, MineolaPérez-Aparicio JL, Palma R, Moreno-Navarro P (2016) Elasto-thermoelectric non-linear, fully coupled, and dynamic finite element analysis of pulsed thermoelectrics. Appl Therm Eng 107:398–409Taylor RL (2010) FEAP a finite element analysis program: user manual. University of California, BerkeleyJiménez JL, Campos I, López-Mariño MA (2013) Maxwell’s equations in material media, momentum balance equations and force densities associated with them. Eur Phys J Plus 128(46):1–6Sardanashvily G (2016) Noether’s theorems. Applications in mechanics and field theory. Springer, BerlinLandau LD, Lifshitz EM (1984) Electrodynamics of continuous media. Pergamon Press Ltd, OxfordJuretschke HJ (1977) Simple derivation of the Maxwell stress tensor and electrostictive effects in crystals. Am J Phys 45(3):277–280Palma R, Pérez-Aparicio JL, Taylor RL (2019) On the non-symmetry of the Maxwell stress tensor: a generalized continuum mechanics approach. Int J Eng Sci (submitted)McMeeking RM, Landis CM (2005) Electrostatic forces and stored energy for deformable dielectric materials. J Appl Mech 72:581–590Reitz JR, Milford FJ, Christy RW (1960) Foundations of electromagnetic theory. Addison-Wesley Publishing Company, Inc, BostonPalma R, Pérez-Aparicio JL, Taylor RL (2017) Dissipative finite element formulation applied to piezoelectric materials with Debye memory. IEEE/ASME Trans Mechatron 23(2):856–863Zienkiewicz OC, Taylor RL, Zhu JZ (2013) The finite element method: its basis and fundamentals, 7th edn. Elsevier, OxfordMoffett MB, Clark AE, Wun-Fogle M, Linberg J, Teter JP, McLaughlin EA (1991) Characterization of Terfenol-D for magnetostrictive transducers. J Acoust Soc Am 89(3):1448–1455Telesnin RV, Shishkov AG (1958) Effect of magnetic viscosity on the frequency properties of ferrites. Sov Phys JETP 6(33):649–652Cole KS, Cole RH (1941) Dispersion and absorption in dielectrics—I alternating current characteristics. J Chem Phys 9(4):341–352Gualdi AJ, Zabotto ML, Garcia D, Bhalla A, Gu R, de Camargo PC, de Oliveira AJ (2016) Understanding the dynamic magnetization process for the magnetoelectric effect in multiferroic composites. J Appl Phys 119(12):411

Dissipative Finite-Element Formulation Applied to Piezoelectric Materials With the Debye Memory

© 2018 IEEE. Personal use of this material is permitted. Permissíon from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertisíng or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.[EN] This work presents a finite-element study of the Debye memory in piezoelectric devices. The memory dependence is due to the spontaneous polarization of the electric dipoles, and it can be understood as a transient viscosity-like effect. The formulation assumes a small strain and rotation hypothesis, and the main contribution is the inclusion of the time-dependent constitutive behavior. For this purpose, a unique numerical formulation that uses convolution integrals is developed to solve the time-dependent electric constitutive equation. A consistent and monolithic finite-element formulation is then obtained and implemented. Finally, a commercial piezoelectric device is simulated for two operational modes, an actuator and a sensor. Several important conclusions on the coupled mechanical and electric fields are reported, and the stability of the time integration scheme is tested by representing the time evolution of the electromechanic energy.Palma, R.; Pérez-Aparicio, JL.; Taylor, RL. (2018). Dissipative Finite-Element Formulation Applied to Piezoelectric Materials With the Debye Memory. IEEE/ASME Transactions on Mechatronics. 23(2):856-863. https://doi.org/10.1109/TMECH.2018.2792308S85686323

Experimental design of dynamic model-based damage identification in piezoelectric ceramics

[EN] A model-based inverse problem strategy is proposed for damage characterization, starting from the electromechanical response measurement as input data, and incorporating a numerical model that simulates the piezoelectric response. Furthermore, a sensitivity analysis is developed to provide a rational basis to correctly design the excitation/measurement system. The model-based inverse problem is solved by minimizing a cost functional using genetic algorithms. The cost functional or discrepancy is defined as the L-2 norm of the difference between experimental and simulated measurements. The latter are obtained by solving the forward problem, using a novel 2D dynamic piezoelectric finite element. The effects of measurement noise and model uncertainties are studied in detail through a sensitivity analysis, where a sensitivity factor is defined and implemented. The proposed inverse problem strategy reconstructs the defect characteristics with sufficient precision, under realistic levels of noise. © 2011 Elsevier Ltd. All rights reserved.The authors would like to thank the Ministerio de Innovacion y Ciencia, Spain, for the FPU Grant AP-2006-02372 and also from Grants Excelencia Junta Andalucia P08-TEP-3641, MCyT DPI 2002-04472-C02-02. Authors would also like to thank Prof. Rafael Gallego for his invaluable contributions to the development of the present work.Rus, G.; Palma, R.; Pérez-Aparicio, JL. (2012). Experimental design of dynamic model-based damage identification in piezoelectric ceramics. Mechanical Systems and Signal Processing. 26:268-293. https://doi.org/10.1016/j.ymssp.2011.06.023S2682932

Non-linear finite element formulation applied to thermoelectric materials under hyperbolic heat conduction model

In the present work, a three-dimensional, dynamic and non-linear finite element to simulate thermoelectric behavior under a hyperbolic heat conduction model is presented. The transport equations, which couple electric and thermal energies by the Seebeck, Peltier and Thomson effects, are analytically obtained through extended non-equilibrium thermodynamics, since the local equilibrium hypothesis is not valid under the hyperbolic model. In addition, unidimensional analytical solutions are obtained to validate the finite element formulation. Numerically, isoparametric eight-node elements with two degrees of freedom (voltage and temperature) per node are used. Non-linearities due to the temperature-dependence on the transport properties and the Joule effects are addressed with the Newton-Raphson algorithm. For the dynamic problem, HHT and Newmark-ß algorithms are compared to obtain accurate results, since numerical oscillations (Gibbs phenomena) are present when the initial boundary conditions are discontinuous. The last algorithm, which is regularized by relating time steps and element sizes, provides the best results. Finally, the finite element implementation is validated, comparing the analytical and the numerical solutions, and a three-dimensional example is presented.This research was partially supported by the Spanish Ministry of Education through Grant No. FPU AP-2006-02372 and also from Grants MICINN BIA-2008-00522, CSD2008-00037 Canfranc Under-ground Physics, Excelencia Junta Andalucia P08-TEP-03641 and "Ayudas Investigacion" from UPV. The authors would also like to thank Prof. Guillermo Rus for his valuable contributions.Palma, R.; Pérez-Aparicio, JL.; Taylor, R. (2012). Non-linear finite element formulation applied to thermoelectric materials under hyperbolic heat conduction model. Computer Methods in Applied Mechanics and Engineering. 213-216:93-103. https://doi.org/10.1016/j.cma.2011.11.011S93103213-21

Elasto-thermoelectric beam formulation for modeling thermoelectric devices

[EN] The present paper provides a dynamic, non-linear and fully coupled Finite Element (FE) formulation based on the Timoshenko beam theory to study elasto-thermoelectric responses in thermoelectric devices. The two main motivations of this work are: i) to study mechanical responses in thermoelectric devices, which must be taken into account in the design of Peltier cells due to the fragility and relative low strength of the semiconductors, and ii) to provide a numerical tool that decreases the CPU time to allow the introduction of designs based on optimization processes and on sensitivity analyses that could require many evaluations. In order to undertake the objectives of this work, the general three-dimensional governing equations are reduced to one-dimensional ones by means of several assumptions. Then, a set of five multi-coupled partial differential equations is obtained. The resultant expressions are thermodynamically consistent and form a multi-coupled monolithic FE formulation, differently to stagger formulations that require two separated steps to reach the final result. Numerically, this set of multi-coupled equations is discretized using the FE method and implemented into FEAP Taylor, 2010 [1]. For a proper validation of the code, four benchmarks are performed using one- dimensional dynamic analytical solutions developed by the authors. Finally, this formulation is compared with a three-dimensional FE formulation also developed by the authors in Pe ́rez-Aparicio et al., 2015 [2] to model a commercial Peltier cell. This comparison reveals that: i) relative errors are lower than 13% and ii) CPU times decrease significantly, more than one order of magnitude. In conclusion, the beam thermoelectric formulation is an accurate model that reduces CPU time and could be used in future design of thermoelectric devices.Palma, R.; Moliner, E.; Pérez-Aparicio, JL. (2017). Elasto-thermoelectric beam formulation for modeling thermoelectric devices. Finite Elements in Analysis and Design. 129:32-41. doi:10.1016/j.finel.2017.02.001S324112

Numerical experiment based on non-linear micropolar finite element to study micro-rotations generated by the non-symmetric Maxwell stress tensor

Funding for open access publishing: Universidad de Granada/CBUA.The main aim of the present work is to investigate the role of the Maxwell stress tensor in the study of active materials. Despite the importance of this tensor in modeling mechatronic devices used in sophisticated applications, its non-symmetry still generates controversies in the literature, probably because classical continuum mechanics assumes a symmetric Cauchy stress, although the sum of Cauchy and Maxwell stresses is non-symmetric. In the framework of generalised continuum mechanics-a more advanced formalism than the classical one-, each material point has an associated microstructure so that the micro-rotations of the electric/magnetic dipoles present in real active materials may be simulated. To this end, a modified total stress formulation, including an angular momentum balance, is developed and implemented into a finite element research code using a complex-step formulation. It is concluded that generalised mechanics allows for incorporating both symmetric and non-symmetric contributions of the Maxwell tensor. Consequently, the rotations generated by the electromagnetic field may be analysed. The influence of the complete Maxwell tensor in a magnetostrictive actuator is studied by several magneto-mechanical numerical experiments of a Terfenol-D rod encircled by air, and several conclusions are highlighted.Universidad de Granada/CBU

Finite element modeling of energy harvesters: application to vibrational devices

[ES] En este capítulo se presenta el conjunto de ecuaciones de gobierno para estudiar el comportamiento de los materiales activos, los cuales tienen una capacidad intrínseca para acoplar varias ramas de la Física y, en consecuencia, son comúnmente utilizados para la fabricación de cosechadoras. Una vez definidas las ecuaciones, se desarrolla una formulación numérica basada en el método de los elementos finitos para modelar estos materiales. En particular, en este capítulo se estudia la producción de energía a partir de las vibraciones mecánicas presentes en puentes ferroviarios de alta velocidad. Para ello, se hace un repaso de los parámetros básicos de estos puentes, sus vibraciones, frecuencias y las características dinámicas. A continuación, se simulan cosechadores en voladizo fabricados con materiales piezoeléctricos y piezomagnéticos bajo vibraciones mecánicas típicas y se destacan varias conclusiones.[EN] This chapter presents the set of governing equations to study the behavior of active materials, which have an intrinsic ability for coupling several branches of Physics and, consequently, are commonly used for manufacturing harvesters. Once the equations are defined, a numerical formulation based on the finite element method is developed in order to model these materials. In particular, this chapter studies the energy production from mechanical vibrations present in high-speed railway bridges. For this purpose, a review of the basic parameters of these bridges, their vibrations, frequencies and the dynamic characteristics are highlighted. Then, cantilever harvesters made out of piezoelectric and piezomagnetic materials are simulated under typical mechanical vibrations and several conclusions are highlighted.Palma, R.; Pérez-Aparicio, JL.; Museros Romero, P. (2018). Finite element modeling of energy harvesters: application to vibrational devices. En Energy Harvesting for Wireless Sensor Networks: Technology, Components and System Design. De Gruyter. 3-33. https://doi.org/10.1515/9783110445053-00133

Optimal measurement setup for damage detection in piezoelectric plates

[EN] An optimization of the excitation-measurement configuration is proposed for the characterization of damage in PZT-4 piezoelectric plates, from a numerical point of view. To perform such an optimization, a numerical method to determine the location and extent of defects in piezoelectric plates is developed by combining the solution of an identification inverse problem, using genetic algorithms and gradient-based methods to minimize a cost functional, and using an optimized finite element code and meshing algorithm. In addition, a semianalytical estimate of the probability of detection is developed and validated, which provides a flexible criterion to optimize the experimental design. The experimental setup is optimized upon several criteria: maximizing the probability of detection against noise effects, ensuring robust search algorithm convergence and increasing the sensitivity to the presence of the defect. The measurement of voltage phi is concluded to provide the highest identifiability, combined with an excitation of the specimen by a mechanical traction transverse to the polarization direction. Sufficient accuracy is predicted for the damage location and sizing under realistic noise levels. (c) 2008 Elsevier Ltd. All rights reserved.This research was supported by the Ministry of Education of Spain through Grant No. FPU AP-2006-02372.Rus, G.; Palma Guerrero, R.; Pérez-Aparicio, JL. (2009). Optimal measurement setup for damage detection in piezoelectric plates. International Journal of Engineering Science. 47(4):554-572. https://doi.org/10.1016/j.ijengsci.2008.09.006S55457247

Plasticity coupled with thermo-electric fields: Thermodynamics framework and finite element method computations

[EN] A consistent thermodynamic-based theoretical framework and three-dimensional finite element formulation is pre- sented, capable of coupling elastic, thermal and electric fields. The complete set of governing equations are obtained from conservation principles for electric charge, energy and momentum. The second principle of thermodynamics is taken into account to introduce the irreversible phenomena, such as plastic dissipation or Joule heating. The constitu- tive relations are derived consistently from the Helmholtz free-energy potential for each corresponding dual variable in terms of the defined set of state variables. We consider the case of linear isotropic hardening model for plasticity, and provide the consistent form of the tangent thermo-electro-elastoplastic modulus through dual variable computa- tions. The latter plays the crucial role in ensuring fast convergence properties of the finite element computations with the proposed coupled plasticity model. The implementation is carried out in a research version of the well-known computer code FEAP. Several numerical simulations are presented in order to illustrate the proposed model and for- mulation capabilities for providing an enhanced formulation of an important practical application in terms of Peltier cells.This work was supported jointly by Haut-de-France Region (CR Picardie) (120-2015-RDISTRUCT-000010 and RDISTRUCT-000010) and EU funding (FEDER) for Chaire-de-Mecanique-Numerique (contract Ref. 120-2015-RDISTRUCTF-000010 and RDISTRUCTI-000004). This support is gratefully acknowledged.Moreno-Navarro, P.; Ibrahimbegovic, A.; Pérez-Aparicio, JL. (2017). Plasticity coupled with thermo-electric fields: Thermodynamics framework and finite element method computations. Computer Methods in Applied Mechanics and Engineering. 315:50-72. doi:10.1016/j.cma.2016.10.038S507231

Linear elastic mechanical system interacting with coupled thermo-electro-magnetic fields

[EN] A fully-coupled thermodynamic-based transient finite element formulation is proposed in this article for electric, magnetic, thermal and mechanic fields interactions limited to the linear case. The governing equations are obtained from conservation principles for both electric and magnetic flux, momentum and energy. A full-interaction among different fields is defined through Helmholtz free-energy potential, which provides that the constitutive equations for corresponding dual variables can be derived consistently. Although the behavior of the material is linear, the coupled interactions with the other fields are not considered limited to the linear case. The implementation is carried out in a research version of the research computer code FEAP by using 8-node isoparametric 3D solid elements. A range of numerical examples are run with the proposed element, from the relatively simple cases of piezoelectric, piezomagnetic, thermoelastic to more complicated combined coupled cases such as piezo-pyro-electric, or piezo-electro-magnetic. In this paper, some of those interactions are illustrated and discussed for a simple geometry.This work was supported jointly by Hauts-de-France Region (CR Picardie) (120-2015-RDISTRUCT-000010 and RDISTRUCT-000010) and EU funding (FEDER) for Chaire-de-Mecanique (120-2015-RDISTRUCTF-000010 and RDISTRUCTI-000004). Also, by the grant Ministerio de Educacion, Cultura y Deporte PRX16/00501. This support is gratefully acknowledged.Moreno-Navarro, P.; Ibrahimbegovich, A.; Pérez-Aparicio, JL. (2018). Linear elastic mechanical system interacting with coupled thermo-electro-magnetic fields. Coupled Systems Mechanics, an international journal (Online). 7(1):5-25. https://doi.org/10.12989/csm.2018.7.1.005S5257